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The paper contains an analysis of the basic criteria for the selection of spreading sequences for the multicarrier CDMA (MC-CDMA) systems with spectrum spreading in the frequency domain. It is shown that the time-domain crosscorrelation function between the spreading sequences is not a proper interference measure for the asynchronous MC-CDMA users. Therefore, the spectral correlation function is introduced and, together with the crest factor and the dynamic range of the corresponding multicarrier waveforms, is used for the evaluation of MC-CDMA sequences. Some well-known classes of sequences, such as Walsh, Gold, Orthogonal Gold, and Zadoff-Chu sequences, as well as Legandre and Golay complementary sequences, are evaluated with respect to the aforementioned basic criteria. It is also shown that the crest factors of the multicarrier spread spectrum waveforms based on the multilevel Huffman sequences are very close to or even lower than the crest factor of a single sine wave.

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Closed-form expressions for the discrete Fourier transform (DFT) of cyclically shifted Zadoff-Chu sequences of arbitrary length are presented. These expressions allow for a very efficient DFT implementation based only on the elements of the corresponding already generated basic (non-cyclically-shifted) ZC sequence.Introduction: Zadoff-Chu (ZC) sequences [1, 2] have been extensively used in various parts of the LTE cellular standard [3]. In particular, the random access (RA) preambles transmitted from the mobile user equipment (UE) are generated from cyclically shifted ZC sequences of a prime length N, by using so-called DFT-S-OFDM modulation ([3], pp. 348 and 438). The first step in generating such RA preambles is to perform an N-point discrete Fourier transform (DFT) of a cyclically shifted ZC sequence characterised by the root index u and the cyclic shift p.Besides this particular application, a closed-form expression of the DFT of ZC sequences is of general interest for any signal design based on the application of ZC sequences in the Fourier frequency domain. An attempt to obtain such a closed-form representation has been made in [4], but the result is valid for a single, very special combination of the sequence length N and the root index u (N even, u ¼ 1). Additionally, this DFT formula does not reveal an efficient DFT implementation structure. Much more general closed-form results, for discrete chirp sequences of arbitrary length, have been obtained in [5]. However, these closed-form expressions still do not reveal the efficient DFT implementation for cyclically shifted ZC sequences.The efficient DFT structures more suitable for applications of ZC sequences have been derived in [6] and [7] for prime length ZC sequences. However, these formulas are semi-analytical, i.e. contain a constant that is supposed to be computed numerically.In this Letter, we present compact closed-form expressions for the DFT of cyclically shifted ZC sequences of arbitrary length as a function based only on the elements of the corresponding already generated basic (non-cyclically-shifted) ZC sequence.

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